Time-Optimal Adaptation in Metabolic Network Models

Analysis of metabolic models using constraint-based optimization has emerged as an important computational technique to elucidate and eventually predict cellular metabolism and growth. In this work, we introduce time-optimal adaptation (TOA), a new constraint-based modeling approach that allows us to evaluate the fastest possible adaptation to a pre-defined cellular state while fulfilling a given set of dynamic and static constraints. TOA falls into the mathematical problem class of time-optimal control problems, and, in its general form, can be broadly applied and thereby extends most existing constraint-based modeling frameworks. Specifically, we introduce a general mathematical framework that captures many existing constraint-based methods and define TOA within this framework. We then exemplify TOA using a coarse-grained self-replicator model and demonstrate that TOA allows us to explain several well-known experimental phenomena that are difficult to explore using existing constraint-based analysis methods. We show that TOA predicts accumulation of storage compounds in constant environments, as well as overshoot uptake metabolism after periods of nutrient scarcity. TOA shows that organisms with internal temporal degrees of freedom, such as storage, can in most environments outperform organisms with a static intracellular composition. Furthermore, TOA reveals that organisms adapted to better growth conditions than present in the environment (“optimists”) typically outperform organisms adapted to poorer growth conditions (“pessimists”).


S1 ABBREVIATIONS AND NOTATIONS
total molecular amounts (of metabolites) within the cell v(t) flux rates in the metabolic network S stoichiometric matrix u(t) collection of (mostly time-dependent) degrees-of-freedom in the network (mostly flux rates) (·) time derivative, e.g.ẏ(t) = d dt y(t) w vector of molecular weights bio(t) biomass w · y(t) of the cell c(t) concentrations of metabolites c = y w ·y 0, 0 zero vector, zero matrix [t 0 , T * ] time interval of interest A(·) (abstract) set defining the constraint-based modeling framework, see (2.7c) λ Growth rate of the cell (given exponential growth from mass m 0 to mass m 1 in a time interval of length ∆: transpose of a vector/matrix (·) i 1 :i 2 indexing of a vector: For a = (a 1 , a 2 , . . . , a n ) and positive integers i 1 ≤ i 2 , it holds a i 1 :i 2 := (a i 1 , a i 1 +1 , . . . , a i 2 ) . If no start/end index is supplied, all remaining indices are included: a i 1 : := (a i 1 , a i 1 +1 , . . . , a n ) ≤ (component-wise) smaller-or-equal relation LP Linear program/linear programming TOA(-VA) time-optimal adaptation (Variability Analysis) FBA flux balance analysis FVA flux variability analysis cFBA conditional FBA RBA resource balance analysis deFBA dynamic enzyme-cost FBA

S2.1 Model Description
Compounds: The vector of metabolic compounds is given by y(t) = (M (t), Tr(t), R(t)) ; additionally, there is an external nutrient supply of total amount N (t) that is controlled by the external conditions.

S2.2 Re-formulation as a deFBA problem
In the terms of Example 2.2, the dynamics of the self-replicator model can be described via:

S2.3 Formulation including Extracellular Compound Dynamics
In the above formulation of TOA, extracellular compounds (like nutrients or waste products) are not explicitly incorporated into the framework and all dynamic evolution (by means of ODEs) is restricted to the total cellular amounts y(t). The influence of the environment to the cell is captured by the abstract constraint set A(t) in the general description (2.3) or, more specifically, by the matrices/vectors H y (t), H v (t), h(t) in (2.4). If the amounts of extracellular compounds (now denoted by e(t) ∈ R ne ) are part of the unknowns in the model, the constraint-based framework (2.3) can be adapted to include dynamic relations, i.e., DAEs includingė(t) as well. Formula (2.3a) can then be extended to for (almost) all t: (ẏ(t),ė(t), y(t), e(t), u(t)) ∈Ã(t) , (2.3a*) with an appropriate extensionÃ(t) of the original constraint set A(t). All frameworks covered by (2.3) can be formulated in terms of (2.3a*) as well and, accordingly, TOA can also be formulated with e(t) as part of the dynamic variables. However, we note that providing goal states for external biochemicals is more unrealistic from the modeling viewpoint as the cell only has very limited control over its surroundings.
Instead of re-formulating the frameworks in Examples 2.1, 2.2, 2.3 and 2.4 in detail, we will just show here (i) how the self-replicator model can be re-framed to include limited total nutrient availability by considering dynamics of N (t) and (ii) how the original model can be recovered again by adapting the flux vector. As the uptake reaction is nonlinear (Michaelis-Menten kinetics), a fully linear model is no longer possible.
and provide initial values for the external nutrient N (t 0 ). In this formulation, no external nutrient source is present anymore and N (t) gets accordingly depleted over time (for TOA, this would have important implications as the environment the cell adapts to is no longer the same after the adaptation process). (ii) If the original problem were to be recovered while still including the dynamics of e(t), one can (there are multiple ways to model this) introduce another flux rate v in (t) ≥ 0 that compensates the reduction of the nutrient. The differential equation for e 1 (t) would then need to be adapted tȯ and the overall nutrient level could be kept at a given level N given (t) via (in-) equality constraints N given (t) ≤ e 1 (t) ≤ N given (t) .
Numerically, this re-formulation is (unnecessarily) complex as the differential equation (A.1) is merely a dummy equation and a linear problem is formulated in nonlinear terms. We state the alternative form here to show that a more general approach using external compounds e(t) is able to capture more general biochemical questions; the above restriction to dynamics of y(t) was mainly for notational convenience.

S3 FORMULATION IN TERMS OF CONCENTRATIONS
The constraints on the boundary values are given by 1. Initial values (as usual): where B y goal ∈ R ny×ny is defined through This means, the boundary conditions to be imposed by the optimal control solver are given by I ny 0 · y(t 0 ) + 0 B y goal · y(t end ) = y (0) 0 |J | .

S4 ENZYME FRACTIONS RELATIVE TO GROWTH RATE
The following plot shows the relation between "optimal" molecular amounts (as predicted by RBA) and growth rate of the cell, cf. Tr R Figure S1. Cellular amounts of intracellular compounds as functions of maximal growth rate. Extracellular nutrient is measured relative to the Michaelis constant K M of the uptake reaction.